LINKER TAXATION
This discussion of the taxation of inflation-indexed bonds is going to be simplistic, especially after the detail of Chapter 4. The stylized P-Linker and C-Linker are assumed to be purchased at par value and held to maturity. There are no capital gains or losses and no de minimis OID, just ordinary income tax. My objective is to demonstrate that these two designs offering inflation protection generate very different after-tax cash flows. Moreover, when the inflation rate is high, the after-tax real rates of return become negative. That explains why linkers usually are held in tax-deferred, retirement portfolios like defined-benefit and defined-contribution pension funds.
Table 7.5 shows the after-tax cash flows on the 2.50%, annual payment, 10-year P-Linker assuming a 30% tax rate on ordinary income and the high-inflation scenario. On date 6, the inflation rate for the year reaches double digits, 15.626%, raising the accrued principal up to $1,613.63 from $1,395.56. The interest payment is $40.34 (= 0.0250 * $1,613.63). The tax obligation on the interest income is $12.10 (= 0.30 * $40.34). But P-Linker taxation does not stop there – the increase in the accrued principal is taxed as ordinary income in the current year even though that compensation for inflation is not received until maturity. This is another example of phantom income. The tax liability on the increase in the accrued principal is $65.42 [= 0.30 * ($1,613.63 – $1,395.56)]. The total tax obligation is $77.52 (= $12.10 + $65.42), resulting in an after-tax cash flow of-$37.18 (= $40.34 – $77.52).
Negative after-tax cash flows for the P-Linker start in the third year and last until maturity in this high-inflation scenario. A useful calculation for the investor is the threshold inflation rate, shown in equation 7.11, which indicates the point at which negative after-tax cash flows arise. It's derived in the Technical Appendix.
TABLE 7.5 After-Tax Cash Flows on the 2.50%, 10-Year, Annual Payment P-Linker, High-Inflation Scenario, 30% Tax Rate
Date |
CPI |
Inflation Rate |
Accrued Principal |
Before-Tax Cash Flow |
Taxes Due |
After-Tax Cash Flow |
After-Tax Real Value |
0 |
100.000 |
1,000.00 |
-1,000.00 |
-1,000.00 |
-1,000.00 |
||
1 |
104.566 |
4.566% |
1,045.66 |
26.14 |
21.54 |
4.60 |
4.40 |
2 |
110.823 |
5.984% |
1,108.23 |
27.71 |
27.08 |
0.62 |
0.56 |
3 |
118.398 |
6.835% |
1,183.98 |
29.60 |
31.60 |
-2.01 |
-1.69 |
4 |
128.005 |
8.114% |
1,280.05 |
32.00 |
38.42 |
-6.42 |
-5.02 |
5 |
139.556 |
9.024% |
1,395.56 |
34.89 |
45.12 |
-10.23 |
-7.33 |
6 |
161.363 |
15.626% |
1,613.63 |
40.34 |
77.52 |
-37.18 |
-23.04 |
7 |
188.857 |
17.039% |
1,888.57 |
47.21 |
96.65 |
-49.43 |
-26.17 |
8 |
211.312 |
11.890% |
2,113.12 |
52.83 |
83.21 |
-30.39 |
-14.38 |
9 |
228.523 |
8.14%5 |
2,285.23 |
57.13 |
68.77 |
-11.64 |
-5.09 |
10 |
240.805 |
5.375% |
2,408.05 |
2,468.25 |
54.91 |
2,413.34 |
1,002.20 |
IRR |
11.814% |
8.362% |
-0.763% |
(7.11)
Fixed Rate is the coupon rate on the P-Linker, here 2.50%, and Tax Rate is the applicable rate on ordinary income, here 30%. Substituting those into equation 7.11 gives a threshold rate of 6.195%.
In general, the lower the fixed coupon rate and the higher the tax rate, the lower is the threshold inflation rate that results in negative after-tax cash flow.
The after-tax cash flows for the 2.50%, annual payment, 10-year C-Linker are shown in Table 7.6 for the same high-inflation scenario. For the sixth year when the inflation rate is 15.626%, the coupon rate is set at 18.126% (= 2.50% + 15.626%) and the interest payment is $181.26 per $1,000 in par value. The tax obligation is $54.38 (=0.30 * $181.26), leaving
TABLE 7.6 After-Tax Cash Flows on the 2.50%, 10-Yeag Annual Payment C-Linker, High-Inflation Scenario, 30% Tax Rate
Date |
CPI |
Inflation Rate |
Coupon Rate |
Before-Tax Cash Flow |
Taxes Due |
After-Tax Cash Flow |
After-Tax Real Value |
0 |
100.000 |
-1,000.00 |
-1,000.00 |
-1,000.00 |
|||
1 |
104.566 |
4.566% |
7.066% |
70.66 |
21.20 |
49.46 |
47.30 |
2 |
110.823 |
5.984% |
8.484% |
84.84 |
24.45 |
59.39 |
53.59 |
3 |
118.398 |
6.835% |
9.335% |
93.35 |
28.01 |
65.35 |
55.19 |
4 |
128.005 |
8.114% |
10.614% |
106.14 |
31.84 |
74.30 |
58.04 |
5 |
139.556 |
9.024% |
11.524% |
115.24 |
34.57 |
80.67 |
57.80 |
6 |
161.363 |
15.626% |
18.126% |
181.26 |
54.38 |
126.88 |
78.63 |
7 |
188.857 |
17.039% |
19.539% |
195.39 |
58.62 |
136.77 |
72.42 |
8 |
211.312 |
11.890% |
14.390% |
143.90 |
43.17 |
100.73 |
47.67 |
9 |
228.523 |
8.145% |
10.645% |
106.45 |
31.93 |
74.51 |
32.61 |
10 |
240.805 |
5.375% |
7.875% |
1,078.75 |
23.62 |
1,055.12 |
438.16 |
IRR |
11.227% |
7.975% |
-0.819% |
an after-tax cash flow of $126.88 (= $181.26 – $54.38). This is much more straightforward than the P-Linker – there is no taxable phantom income or negative after-tax cash flow.
The sad news is that both of these linkers are projected to deliver negative after-tax real IRRs. To be sure, the realized real rates of return will depend on actual real rates when the coupons are reinvested. These results ultimately depend on the particular price and rate assumptions. It's easy to put these stylized linkers onto a spreadsheet to see the impact of lowering the purchase price, raising the fixed coupon rate, lowering the tax rate, and lowering the average inflation rate. Those changes raise the after-tax real IRR and can make it a positive outcome. For instance, other things being equal, if the tax rate is less than 23.04% on this 2.50%, 10-year P-Linker, and less than 22.21% on the C-Linker, the after-tax IRRs are above zero. An individual investor can manage the tax problem by holding the linker in a tax-deferred, retirement savings account like a 401(k) or 403(b). Doing so won't make the tax obligation go away, but will allow the investment to compound at the before-tax real yield.
Yield duration in Chapter 6 is defined as the sensitivity of the fixed-income bond price to a change in the nominal yield to maturity. Inflation-indexed bonds require that we focus on why the nominal rate changes and distinguish between a change in the real rate and a change in the inflation rate. Let's start by formalizing the stylized linkers, keeping close to the notation of Chapter 3. Let the nominal rate be y, the real rate r, and the inflation rate i. Also, let the number of periods to maturity be N, the fixed coupon rate c, the par (or face) value of the linker FV, and the current price PV. For these stylized linkers, we are on a coupon payment date so there is no accrued interest to sully the equations.
P-Linker valuation is based on the assumed path for the accrued principal. Given a constant inflation rate, this path will be (1 + i) * FV, (1 + i)2 * PV, ..., (1 + i)N * PV. Then the price of the P-Linker, denoted PVPLINK, is the present value of the cash flows, discounted at the nominal rate.